Effects of spin vacancies on magnetic properties of the Kitaev-Heisenberg model

We study the ground-state properties of the Kitaev-Heisenberg model in a magnetic field and explore the evolution of spin correlations in the presence of nonmagnetic vacancies. By means of exact diagonalizations, the phase diagram without vacancies is determined as a function of the magnetic field and the ratio between Kitaev and Heisenberg interactions. We show that, in the (antiferromagnetic) stripe-ordered phase, the static susceptibility and its anisotropy can be described by a spin canting mechanism. This accounts as well for the transition to the polarized phase when including quantum fluctuations perturbatively. Effects of spin vacancies depend sensitively on the type of the ground state. In the liquid phase, the magnetization pattern around a single vacancy in a small field is determined, and its spatial anisotropy is related to that of nonzero further neighbor correlations induced by the field and/or Heisenberg interactions. In the stripe phase, the joint effect of a vacancy and a small field breaks the sixfold symmetry of the model and stabilizes a particular stripe pattern. Similar symmetry-breaking effects occur even at zero field due to effective interactions between vacancies. This selection mechanism and intrinsic randomness of vacancy positions may lead to spin-glass behavior.

Figure 1

(a) Schematic picture of an octahedron (6 oxygens surrounding a Ir${}^{4+}$ ion) with the octahedral axes $x$, $y$, $z$, which are easy axes on identically labeled bonds in the Kitaev-Heisenberg model. (b) A hexagon in the Ir${}^{4+}$ plane, indicating bonds labeled $x$, $y$, $z$ according to their local easy axis regarding Kitaev interactions. (c) Honeycomb (direct) lattice with the labeling $x,y,z$ of bonds according to their in-plane orientation. A unit cell (centered on a $z$ bond) is represented in the dashed rectangle, as well as elementary translations ${\vec{n}}_{1/2}$. (d) First Brillouin zone, with the position of stripe-order wave vectors ${\vec{Q}}_{\gamma }$ ($\gamma =x,y,z$).

Figure 2

(a) Representation of a $z$-stripe pattern (stripe pattern with spins oriented along the $z=\left[001\right]$ axis-neighboring spins are aligned on $z$ bonds and antialigned on other bonds). (b) Schematic representation, at the level of a single hexagon, of a canted stripe pattern: here, a $z$-stripe pattern, with spins tilted of an angle $\theta$ toward a field along the [100] direction.

Figure 3

Magnetization $m_0$ (per site) and $x$-stripe structure factor $S^x\left({\vec{Q}}_x\right)$ versus $h_z$ for various values of $\alpha$ and $N$. Dotted lines: Perturbative results for $m_0$ in the polarized phase. Inset: Phase diagram in the $(\alpha ,h_z)$ plane; continuous transition lines are interpolations of points from $N=24$ data and dashed lines are perturbative estimates.

Figure 4

Susceptibility per spin ${\chi }^z$, evaluated as $m_0/h_z$ for small $h_z=0.01$, and its equivalent ${\chi }^{\left[111\right]}$ for a $\left[111\right]$-oriented field, on clusters $N=24$ or 32. Shown also is the quantity $4(1-\alpha ){\chi }^z$ for $N=32$.

Figure 5

Evolution of correlations $|⟨{\sigma }_i^z{\sigma }_j^z⟩|$ with $\alpha$ on a $N=24$ cluster at $h=0$ for various relative positions $(i-j)$, with $i$ and $j$ indicated in the inset. The (red) dot indicates the value $\sim 0.53$ of the $(0-1)$ correlation at $\alpha =1$, which coincides with an analytical result from Ref. 14.

Figure 6

(a), (b) Correlation functions $⟨{\sigma }_i^z{\sigma }_j^z⟩$ with respect to site $i$ indicated by a black circle. (c), (d) Induced magnetization patterns $\delta {\sigma }_j^z$ due to the presence of a vacancy on the $N=24$ cluster. Circle areas are proportional to the quantities shown. Parameters are, respectively, $(\alpha ,h_z)=(1,0.05)$ in (a) and (c), while $(\alpha ,h_z)=(0.9,0)$ in (b) and (d) [a small symmetry-breaking field $h_z=\epsilon ={5.10}^{-4}$ was added in (d)].

Figure 7

Spin-vacancy-induced magnetization $\delta M^z$ as a function of magnetic field $h_z$ for different $\alpha$ and cluster sizes. Shown also is a partial contribution (open circles) of the spin at site $k$, being the $z$ neighbor of the vacancy for $\alpha =1$ and $N=24$.

Figure 8

The $z$-stripe order parameter $m^z\left({\vec{Q}}_z\right)$ computed on the $N=24$ cluster as a function of $\alpha$ at different small fields $h_z$. Right inset: The magnetization pattern $⟨{\sigma }_{\vec{r}}^z⟩$ around the vacancy for $\alpha =0.7$ and $h_z={10}^{-3}$. Left inset: Total magnetization $M^z$ as a function of a field $h_z$ applied in the liquid ($\alpha =0.9$) and in the stripe ($\alpha =0.7$) phases, with a magnetization jump at $h_{an}\simeq 0.062$.

Figure 9

Magnetization curves $M^z\left(h_z\right)$ on the cluster $N=24$ with 0, 1, or 2 vacancies. The labels P1, … ,P4 correspond to relative positions of two vacancies as shown in panel (c). Panels (a), (b), and (c) correspond to $\alpha =0.1$, 0.7, and 0.9, respectively.

Figure 10

Structure factors (a) $S^z\left({\vec{Q}}_z\right)$ and (b) $S^x\left({\vec{Q}}_x\right)$ at $h=0$ calculated for various respective positions of two vacancies P1, P2, and P3 indicated in the insets. Also shown is the structure factor of the vacancy-free system (open circles) for which $S^z\left({\vec{Q}}_z\right)=S^x\left({\vec{Q}}_x\right)$. All results are obtained on the $N=24$ cluster.